Global exponential stability and existence of almost periodic solutions in distribution for Cli ﬀ ord-valued stochastic high-order Hopﬁeld neural networks with time-varying delays

: In this paper, we consider a class of Cli ﬀ ord-valued stochastic high-order Hopﬁeld neural networks with time-varying delays whose coe ﬃ cients are Cli ﬀ ord numbers except the time delays. Based on the Banach ﬁxed point theorem and inequality techniques, we obtain the existence and global exponential stability of almost periodic solutions in distribution of this class of neural networks. Even if the considered neural networks degenerate into real-valued, complex-valued and quaternion-valued ones, our results are new. Finally, we use a numerical example and its computer simulation to illustrate the validity and feasibility of our theoretical results.


Introduction
Clifford-valued neural networks (NNs) are the NNs whose state variables, connection weights and external inputs are Clifford numbers. They are generalizations of real-valued, complex-valued and quaternion-valued neural networks. In recent years, due to their advantages over real-valued networks and their potential application values in many fields, they have attracted the attention of many researchers [1][2][3][4][5][6][7][8][9][10][11][12]. However, because the multiplication of Clifford numbers does not satisfy the commutative law, it is difficult to study the dynamics of Clifford-valued NNs. At this stage, there are few results on the dynamics of Clifford-valued NNs [8][9][10][11][12][13][14]. In addition, it is worth noting that in most of the existing results [5,[7][8][9]14], the coefficients of the leakage terms in neural networks are assumed to be real numbers.
On the one hand, it is well known that high-order Hopfield NNs have more advantages than loworder Hopfield NNs. Therefore, in the past few decades, many scholars have done a lot of research on the dynamics of high-order Hopfield NN [13,[15][16][17][18]. This is because the application of neural networks in various fields largely depends on their dynamic performance. Moreover, the use of neural networks with complex or even chaotic dynamic behaviors in information processing is expected to improve the efficiency and flexibility of information processing.
In addition, noise interference is the main source of neural network instability, which can lead to poor neural network performance. In the real nervous system, synaptic transmission is a noisy process, caused by random fluctuations in neurotransmitter release and other probabilistic reasons. As we all know, neural networks can be stable or unstable through some random inputs [19]. For this reason, stochastic neural networks are widely studied [20][21][22][23][24][25][26].
Besides, we know that the existence and stability of equilibrium points are important dynamics of autonomous neural networks. For nonautonomous neural networks, there are generally no equilibrium points. Therefore, the existence and stability of periodic or almost periodic solutions are important dynamics. Since almost periodicity is more common than periodicity, in the past few decades, many scholars have studied the almost periodic solutions of deterministic neural networks [7,[27][28][29]. However, the existing results on the existence of almost periodic solutions of stochastic neural networks are almost all about mean-square almost periodic solutions. Unfortunately, in [30], some counterexamples show that the nontrivial solutions of some stochastic differential equations with almost periodic coefficients cannot be mean-square almost periodic. Therefore, it is more reasonable to study the almost periodic solutions in distribution of stochastic differential equations. Random almost periodic oscillation is a complex oscillation phenomenon. However, so far, no papers have been published on almost periodic solutions in distribution of Clifford-valued stochastic high-order Hopfield NNs. Therefore, it is necessary to study this issue.
Inspired by the above discussion, and considering the fact that time delay is inevitable, in this work, we consider the following Clifford-valued stochastic high-order Hopfield NN with time varying-delays: are the activation functions of signal transmission; ω(t) = (ω 1 (t), ω 2 (t), . . . , ω n (t)) T is an n-dimensional Brownian motion defined on a complete probability space; δ pq : A → A is a Borel measurable function. Let (Ω, F , {F t } t≥0 , P) be a complete probability space with a natural filtration {F t } t≥0 satisfying the usual conditions. Denote by The initial values of system (1.1) are given by The main purpose of this paper is to study the existence and global exponential stability of almost periodic solutions in distribution of system (1.1). The innovations of this paper are as follows: (1) This is the first paper that uses a non-decomposition method to study stochastic NNs whose coefficients are all Clifford numbers except for time delays. (2) This is the first time to study almost periodic solutions in distribution of Clifford-valued stochastic high-order Hopfield NNs. (3) The method of dealing with time-varying delays in this paper can be used to study the corresponding problems of other types of stochastic NNs with time-varying delays. (4) When the system we consider degenerates into realvalued system, complex-valued system or quaternion-valued system, the results of this paper are also new.
The rest of this paper is organized as follows. In Sect. 2, we recollect some basic definitions and lemmas. In Sect. 3, based on the principle of contractive mapping, we establish the existence of almost periodic solutions in distribution for system (1.1). In Sect. 4, we study the global exponential stability of the almost periodic solution in distribution of system (1.1) by inequality techniques. In Sect. 5, we give an example to illustrate the feasibility of the theoretical results obtained in this paper. In Sect. 6, we give a concise conclusion to end this paper.
Throughout this paper, for The τ is called the ε-translation number of f . Denote by AP(R, A n ) the set of all such functions.
Let (Ω, F , {F t } t≥0 , P) be a complete probability space with a natural filtration {F t } t≥0 satisfying the usual conditions.
A stochastic process Let (E, d) be a separable, complete metric space and B(E) be the σ-algebra of Borel sets of E. We denote by P(E) the set of all probability measures defined on B(E) and by CB(E) the set of all bounded continuous functions f : For f ∈ CB(E), µ, ν ∈ P(E), we define It is well known that the metric space (P(E), d BL ) is a Polish space [32]. For a random variable X : (Ω, F , P) → E, we will denote by µ(X) := P • X −1 its law and by E(X) its expectation.
Let L 2 (Ω, A n ) be the space of all A n -valued random variables such that For X ∈ L 2 (Ω, A n ), we denote X 2 n = Ω X 2 n dP 1 2 and E X 2 n = Ω X 2 n dP.
[34] An L 2 -continuous stochastic process X : R → L 2 (Ω, A n ) is said to be squaremean almost periodic if for every ε > 0, there exist a positive number such that every interval of length contains a number τ such that Definition 2.4. [35] A stochastic process X : R → A n is said to be almost periodic in distribution if the mapping t → µ t := µ(X(t)) is almost periodic, where µ(X(t)) = P • [X(t)] −1 is the law of X(t) under P, that is to say, if for every ε > 0, there exists a positive number such that every interval of length contains a number τ such that From Remark 2.12 in [33], one can deduce that Lemma 2.1. If an L 2 -continuous stochastic process X(t) is square-mean almost periodic, then X(t) is almost periodic in distribution; but the converse is not true.
[36] Let g : R → R be a continuous function such that, for every t ∈ R, where α, β, δ ≥ 0 are constants and δ > β. Then we have g(t) ≤ α δ δ−β . In the rest part of this paper, we will adopt the following notation: Throughout this paper, we assume that

The existence of almost periodic solutions in distribution
We denote by UCB(R, L 2 (Ω, A n )) the space of all L 2 -bounded and uniformly L 2 -continuous Definition 3.1. An F t -progressively measurable stochastic process x(t) is called a mild solution of system (1.1), if x(t) satisfies the following stochastic integral equation Theorem 3.1. Assume that (H 1 )-(H 4 ) hold, then system (1.1) has a unique almost periodic solution in distribution in the closed ball Proof. According to Definition 3.1, taking the limit as t 0 −→ −∞, we obtain which is a mild solution of system (1.1). Define an operator Φ : where for t ∈ R, p ∈ D, Firstly, we will prove that the operator Φ is well defined. and Φx By the Cauchy-Schwarz inequality, we have and Similarly, we have Moreover, by the Itô isometry, we obtain For any x ∈ B κ and t 1 , t 2 ∈ R with t 1 > t 2 , we derive that Consequently, we deduce that E (Φx)(t 1 ) − (Φx)(t 2 ) 2 0 → 0 as t 1 → t 2 , which implies Φx is uniformly L 2 -continuous. Therefore, we gain Φ(B κ ) ⊂ B κ , that is, Φ is well defined.
Next, we will show that Φ is a contraction operator. Actually, for every x, y ∈ B κ , one has Hence, which implies that Φ is a contraction mapping. Therefore, Φ has a unique fixed point x in B κ , that is, system (1.1) possesses a unique solution x in B κ . Finally, we will show that the unique solution x of system (1.1) in B κ is almost periodic in distribution.
Then b pql (s + ς) g q (x q (s + ς − σ pql (s + ς))) × g l (x l (s + ς − ν pql (s + ς))) − g q (x q (s − σ pql (s)))g l (x l (s − ν pql (s))) ds According to the Cauchy-Schwarz inequality, we have that Similarly, we can get 14) Note that By a same method, one gets Similar to (3.18) and by the Itô isometry, one has and Substituting (3.9)-(3.21) into (3.8), we get Thus, by Lemma 2.2, we conclude that which implies that x(t) is square-mean almost periodic. According to Lemma 2.1, we deduce that x(t) is almost periodic in distribution. The proof is complete.

Global exponential stability in mean square
Definition 4.1.
[37] Let x be an almost periodic solution in distribution of (1.1) with the initial value ϕ. If there exist positive constants λ > 0 and M > 0 such that for every solution y with initial value ψ satisfies n , then the almost periodic solution x(t) in distribution of (1.1) is said to be globally exponentially stable. Proof. From Theorem 3.1, we know that system (1.1) has a unique almost periodic solution x in distribution with the initial value ϕ. Suppose that y is an arbitrary solution of (1.1) with initial value ψ. Set z = x − y, then from (1.1), we get Let Υ p : R → R be defined as follows: By (H 4 ), we have Υ p (0) > 0. Due to the continuity of Υ p on [0, +∞) and the fact that Υ p ( ) → −∞, as → +∞, there exists ς p > 0 such that Υ p (ς p ) = 0 and Υ p ( ) > 0 for ∈ (0, ς p ). Consequently, one can take a positive constant 0 < λ < min 1≤p≤n {ς p , c ∅ p } such that Υ(λ) > 0, p ∈ D, which implies that then by (H 4 ), we know M > 1, and further, we can deduce that Hence, for any > 0, it is easy to see that We assert that On the contrary, there exists a certain t 1 > 0 such that Multiplying (4.1) by e s 0 c ∅ p (u)du and integrating it over the interval [0, t], we deduce that, for p ∈ D, − g q (y q (s − σ pql (s)))g l (y l (s − ν pql (s))) ds δ pq (x q (s − γ pq (s))) − δ pq (y q (s − γ pq (s))) dω q (s).

Conclusions
In this work, we use a direct method to prove the existence and global exponential stability of almost periodic solutions in distribution for Clifford-valued stochastic higher-order Hopfield neural networks with all parameters being Clifford numbers except time delays. Even when the neural network considered in this paper degenerates into real-valued, complex-valued and quaternion-valued neural networks, our results are new. In addition, the method proposed in this paper can be used to study the almost periodic solutions in distribution for other types of Clifford-valued stochastic neural networks with time-varying delays.